Question: All of the 4th grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$3.50$ each for students, and the group paid $$50.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$22.50$ each for teachers and $$13.00$ each for students, and the group paid $$175.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+3.5y = 50}$ ${22.5x+13y = 175}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-22.5x-10.5y = -150}$ ${22.5x+13y = 175}$ Add the top and bottom equations together. $ 2.5y = 25 $ $ y = \dfrac{25}{2.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {7.5x+3.5y = 50}$ to find $x$ ${7.5x + 3.5}{(10)}{= 50}$ $7.5x+35 = 50$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {22.5x+13y = 175}$ and get the same answer for $x$ ${22.5x + 13}{(10)}{= 175}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.